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Theorem nfsum1 11093
Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypothesis
Ref Expression
nfsum1.1  |-  F/_ k A
Assertion
Ref Expression
nfsum1  |-  F/_ k sum_ k  e.  A  B

Proof of Theorem nfsum1
Dummy variables  f  j  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sumdc 11091 . 2  |-  sum_ k  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
) ) ) )
2 nfcv 2258 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2258 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3060 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
63nfcri 2252 . . . . . . . 8  |-  F/ k  j  e.  A
76nfdc 1622 . . . . . . 7  |-  F/ kDECID  j  e.  A
84, 7nfralxy 2448 . . . . . 6  |-  F/ k A. j  e.  (
ZZ>= `  m )DECID  j  e.  A
9 nfcv 2258 . . . . . . . 8  |-  F/_ k
m
10 nfcv 2258 . . . . . . . 8  |-  F/_ k  +
113nfcri 2252 . . . . . . . . . 10  |-  F/ k  n  e.  A
12 nfcsb1v 3005 . . . . . . . . . 10  |-  F/_ k [_ n  /  k ]_ B
13 nfcv 2258 . . . . . . . . . 10  |-  F/_ k
0
1411, 12, 13nfif 3470 . . . . . . . . 9  |-  F/_ k if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
152, 14nfmpt 3990 . . . . . . . 8  |-  F/_ k
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
169, 10, 15nfseq 10196 . . . . . . 7  |-  F/_ k  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
17 nfcv 2258 . . . . . . 7  |-  F/_ k  ~~>
18 nfcv 2258 . . . . . . 7  |-  F/_ k
x
1916, 17, 18nfbr 3944 . . . . . 6  |-  F/ k  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x
205, 8, 19nf3an 1530 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )
212, 20nfrexya 2451 . . . 4  |-  F/ k E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )
22 nfcv 2258 . . . . 5  |-  F/_ k NN
23 nfcv 2258 . . . . . . . 8  |-  F/_ k
f
24 nfcv 2258 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
2523, 24, 3nff1o 5333 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
26 nfcv 2258 . . . . . . . . . 10  |-  F/_ k
1
27 nfv 1493 . . . . . . . . . . . 12  |-  F/ k  n  <_  m
28 nfcsb1v 3005 . . . . . . . . . . . 12  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
2927, 28, 13nfif 3470 . . . . . . . . . . 11  |-  F/_ k if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 )
3022, 29nfmpt 3990 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )
3126, 10, 30nfseq 10196 . . . . . . . . 9  |-  F/_ k  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) )
3231, 9nffv 5399 . . . . . . . 8  |-  F/_ k
(  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
)
3332nfeq2 2270 . . . . . . 7  |-  F/ k  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
)
3425, 33nfan 1529 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
) )
3534nfex 1601 . . . . 5  |-  F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
) )
3622, 35nfrexya 2451 . . . 4  |-  F/ k E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
) )
3721, 36nfor 1538 . . 3  |-  F/ k ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  (
ZZ>= `  m )DECID  j  e.  A  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
) ) )
3837nfiotaxy 5062 . 2  |-  F/_ k
( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
) ) ) )
391, 38nfcxfr 2255 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 682  DECID wdc 804    /\ w3a 947    = wceq 1316   E.wex 1453    e. wcel 1465   F/_wnfc 2245   A.wral 2393   E.wrex 2394   [_csb 2975    C_ wss 3041   ifcif 3444   class class class wbr 3899    |-> cmpt 3959   iotacio 5056   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742   0cc0 7588   1c1 7589    + caddc 7591    <_ cle 7769   NNcn 8688   ZZcz 9022   ZZ>=cuz 9294   ...cfz 9758    seqcseq 10186    ~~> cli 11015   sum_csu 11090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-recs 6170  df-frec 6256  df-seqfrec 10187  df-sumdc 11091
This theorem is referenced by:  mertenslem2  11273
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